Coupled square well model and Fano-phase correspondence

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by  Bin Yan

Coupled square well model and Fano-phase correspondence
Bin Yan and Chris H. Greene∗
Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
(Dated: January 11, 2017)
This paper investigates the Fano-Feshbach resonance with a two-channel coupled-square-well
modelinboththefrequencyandtimedomains. ThissystemsisshowntoexhibitFanolineshapepro-
filesintheenergyabsorptionspectrum. Theassociatedtime-dependentdipoleresponsehasaphase
shiftthathasrecentlybeenunderstoodtoberelatedtotheFanolineshapeasymmetricqparameter
7 byϕ=2arg(q−i). Thepresentstudydemonstratesthatthephase-q correspondenceisgeneral for
1 any Fanoresonance in theweak coupling regime, independentof thetransition mechanism.
0
2 PACSnumbers: 32.80.Qk,07.60.Rd,32.70.-n,33.80.Eh

n
a
J I. INTRODUCTION far beyond the area of atomic physics, e.g., condensed
0 mattersystems[26,27],plasmasystems[28,29],highen-
1 Understandingtime-dependentquantumdynamicshas ergyprocesses[30],oroptomechanicssystems[31]. Thus,
emerged as one of the fundamental problems in physics it would be more interesting to have a unified treatment
] of this phase-q relation for any Fano resonance.
h [1]. In recent years, with the development of new tech-
p nologies, especially with ultrashort light sources and ul-
- trafast optical techniques, it has become possible to ex-
m
perimentallyprobetherealtimeelectrondynamicsinthe
In the present work, with all these questions in mind,
o quantum regime [2–9], e.g., time-domain measurements
we focus on an analytically solvable two-channel square
at of the autoionization dynamics using attosecond pulses well model and study its resonance physics in both the
. [10–12], creation and control of time-dependent electron
s frequency and time domains. As an extension of a text-
c wave packet [13, 14]. Studying time domain resonance book single-channel square-well scattering problem, the
i physicshasbeenattractingincreasinginterestsinatomic
s coupled-channels model captures much of the physics of
y and molecular physics [15–20].
near-threshold bound and scattering states [32]. This
h A recent study [21] has both theoretically established
modelhasbeenusedtosuccessfullyexplainthethreshold
p andexperimentallyverifiedageneralcorrespondencebe-
scattering of cold neutrons from atomic nuclei [33] and
[
tween the photon absorption lineshape in the frequency to representFeshbachresonancesin ultracoldatomscat-

1 domain,whichischaracterizedbyaFanolineshapeasym- tering processes[34–36]. Investigationofthe Fano-phase
v metryparameterq [22],andthephaseshiftϕofitstime-
correspondence with such a model would then general-
0 dependent dipole response:
ize the previous result in the dielectric atomic systems
4
6 to a more general class of scenarios, and thus extend its
ϕ=2arg(q−i). (1)
2 potential applications.
0
In a further development, it was shown that by coupling
.
1 the system with a short pulsed laser immediately after
0 the excitation, the phase ϕ of its dipole response can be This paper is organized as following. Section II in-
7
externally controlled. In this way, the q-parameter of troducesthe two-channelcoupled-square-wellmodel. By
1
: the system’s subsequent absorption spectrum can be ef- adding an auxiliary ground state belonging to a third
v fectively modified. In the frequency domain, the Fano channel, we study the energy absorptionspectrum when
i
X q parameter provides a sensitive test of atomic struc- the system is excited from the ground state to the cou-
r turecalculationsunderfield-freeconditions[23,24]. The pled two channels through magnetic dipole transitions.
a phase-q relation thus provides a possible way to control AstandardFanolineshapeisobservedfortheabsorption
aspects of the time-dependent quantum dynamics. cross section, with the asymmetry q parameter linearly
The above infrared laser pulse control mechanism of depending on the transition dipole ratio d /d , consis-
2 1
thephaseshiftϕwasexplainedbybothaquasi-classical, tentwithFano’sconfigurationinteractiontheory[22]. In
ponderomotive-motion picture [21] and in terms of res- section III, the dipole response in the time domain is
onant coupling dynamics [25]. However, the universal studied. The phase-q correspondence Eq. (1) is revealed
phase-q correspondence Eq.(1) was only demonstrated numerically in the present model problem for magnetic
as a generalmacroscopicproperty of a dielectric system, dipole transitions. A general proof of this relation for
thoughithasalreadybeenfaithfullyappliedtoscenarios anytransitionmechanismisalsopresented. Finally,Sec.
IV summarizes our conclusions. Derivation of the eigen
solutions and discussions of the scattering properties of
the two-channel square-well model are given in the Ap-
∗ [email protected] pendix.
2
II. COUPLED SQUARE WELL MODEL
The two-channel square well model in the present
study describes two particles with reduced mass m in-
teracting in three dimensions with the following s-wave
Hamiltonian in the relative coordinate r:
¯h2 d2
Hˆ =− Iˆ +Vˆ(r)+Eˆth. (2)
2m dr2
Here the potential coupling matrix is assumed to vanish
at r >r , but a constant 2×2 matrix at r <r :
0 0
−V V
Vˆ(r)= 1 12 θ(r−r ). (3)
(cid:20)V12 −V2(cid:21) 0
FIG. 1. Resonance profiles at various transition dipoles are
Wearemostinterestedinthecaseforwhichthediagonal showns as cross sections versus the energy. The solid blue
linesarethenumericallycalculatedabsorptioncrosssections.
elements areattractive,whichis why anegativesignhas
Dashed red lines are standard Fano lineshapes (Eq.(9)) with
beenseparatedoutfromthisequationattheoutset,given
that V1 and V2 are positive. The matrix Eˆth containing aǫr=ba1c.6k5grao.uu.ndancdroresssonseacntcioenwiσd0t=h1Γ.9=E-14.,7Ere−so2n.aMncoedeplopsoittieonn-
the real energy thresholds is diagonal. We choose the tialparametersarefixedatV1=75,V2 =10,V12 =10,E2th=
lower threshold, channel |1i in our notation, as defining
2,r0 = 3. The transition dipole parameter d1 is fixed to be
the zero of our total energy scale, whereby 1. The corresponding Fano lineshape parameters and values
of d2 are a) d2 = −0.137,q = 0; b) d2 = −0.130,q = 1; c)
Eˆth = 0 0 . (4) d2=−0.144,q=−1and d) d2 =0.600,q=100.
(cid:20)0 Eth(cid:21)
2
Thismodelhasasingleanalyticsolutionbetweenthetwo where γ is a dimensionless interaction strength param-
energy thresholds, which contains both an exponentially eter. dˆ = d |1ih0| + d |2ih0| is the transition dipole.
1 2
decayingsolutioninthe closedchannel|1ianda scatter- Parameters d and d control the transition strengths
1 2
ing solution in the open channel |2i. In matrix form the into the corresponding channels.
1 0
channel bases read |1i = and |2i = . The radial
(cid:20)0(cid:21) (cid:20)1(cid:21) The wavefunction immediately after the excitation
parts of the energy eigenfunctions are linear combina- pulse is given by
tions of the two channels’ configuration basis functions:
|ǫi = φ1(r;ǫ)|1i+φ2(r;ǫ)|2i, the derivation of which is |ψ(t=0)i=e−iR00−+dtHˆδt|gi
presented in the Appendix. (6)
For appropriate potential parameters, there exist one =e−iγdˆ|gi.
or more bound states below the lower energy threshold.
In the perturbative limit, where γ ≪ 1, the evolution
Up to this point, external field excited transition from
operator is expanded to the first order of γ:
these bound states can then be investigated. However,
in order to simplify the model without losing the key |ψ(t=0)i≈|gi−iγdˆ|gi
features of the problem, while making it extendable to (7)
problemsinvolvingmorethantwochannels,wemodelthe =|gi−iγ dǫhǫ|dˆ|gi|ǫi,
Z
ground state with an auxiliary channel |0i independent
with channel |1i and |2i. This might correspond to an where |ǫi denotes the energy eigenstates. The photo ab-
independent degree of freedom in realistic systems, such sorption cross section can then be calculated as
as hyperfine spin state in a cold atom pair.
σ(ǫ)=|hψ|ǫi|2
Suppose initially the system is prepared in the s-wave (8)
ground state |gi=f(r)|0i, where f(r) is the normalized =γ2|hǫ|dˆ|gi|2.
radial part of the ground state wave function in the co-
According to Fano’s configuration interaction theory,
ordinate representation. In this paper, we consider the
in the energy range between the two threshold energies,
groundstatetobestronglylocalized,andfordefiniteness
where the bound states in the first channel are coupled
we take f(r)=2e−2r. At t=0, a strong δ pulse couples
to the continuum of the second channel, the resonance
the auxiliary channel |0i to the two channels through
profile at each resonance point is predicted to have a
magnetic dipole interaction, and then consequently ex-
simple form:
cites the system to channel |1i and |2i. The short pulse
in modeled by a delta interaction,
(q+¯ǫ)2
Hˆδ =γδ(t)dˆ+h.c., (5) σFano(ǫ)=σ0 1+ǫ¯2 , (9)
3
where ǫ¯ = ǫ−ǫr. With the assumption of a flat-
Γ/2
backgroundnearresonanceandconstantcouplingpoten-
tial V, the q parameter is defined by
hα|dˆ|gi
q ≡ , (10)
πVhβ |dˆ|gi
E
where |αi and |β i are respectively the bare closed-
E
channel bound state and the unperturbed open-channel
energy-normalized continuum eigenstate. In the present
model problem, for fixedsystem potential parameters,it
can be further deduced that
d2
q ∝ . (11)
d1 FIG.2. qversusd2/d1. Dotsarenumericallyfittedqparame-
ters,whichexhibitalineardependenceond2/d1,asexpected
In our numerical study, we tune the potential parame-
from Eq. (11)
ters such that there is exactly one bound state in the
firstchannel,andsuchthatthe backgroundcrosssection
is relatively flat near the position of the resonance. In
Fig. (1) the cross sections for different transition prob-
abilities are plotted. The Fano and Lorentz line profiles
can both be realized by tuning the ratio between d and
2
d . Fig. (2)showsthenumericallyfittedq parametersat
1
variousvaluesofd /d ,whichmatchesthelinearrelation
2 1
as predicted by Eq. (11).
III. DIPOLE RESPONSE
Withtheaidofthetimedependentwavefunctionafter
the strong short pulse excitation,
FIG. 3. Time-dependent dipole response for different q-
parameters. The model potential parameters are chosen as
|ψ(t)i=e−iǫgt|gi−iγ dǫe−iǫthǫ|dˆ|gi|ǫi, (12) in Fig. 1. The solid blue line, dotted red line and dashed
Z
green line correspond to q=0,1,2 respectively.
the dipole response function can be calculated as the
quantum averageof the dipole transition operator:
We note the following remarks: 1) The above deriva-
d(t)=hψ(t)|dˆ|ψ(t)i tion involves only the physically measurable real quan-
tities d(t) and σ(ǫ), and is general for any transition in-
=2Re[iγZ dǫhg|dˆ†|ǫihǫ|dˆ|giei(ǫ−ǫg)t] (13) teraction and model Hamiltonian, as long as wavefunc-
tion Eq.(12) is valid in the perturbative limit. This gen-
2
= Im[ dǫσ(ǫ)e−i(ǫ−ǫg)t], eralizes the application of the phase-q correspondence
γ Z
Eq.(1), which was originally developed in Ref. [21]
where Re and Im denote the real part and imaginary for macroscopic dielectric systems; in that study, the
part, respectively. In the case of a Fano resonance with complex dipole in the energy domain d˜(ǫ) and relation
cross section Eq.(9), the above shifted Fourier transform σ(ǫ) ∝ Im[d˜(ǫ)] [37] were used as the starting point. 2)
can be evaluated directly: Thefrequencyofthedipoleresponseisofcoursethetran-
sitionenergybetweenthegroundstateandtheresonance
dǫσ (ǫ)e−i(ǫ−ǫg)t =2πσ δ(t) energy (neglecting the small resonance level shift due
Fano 0
Z (14) to discrete-continuum level mixing), in agreement with
+πσ0(Γ/2)e−Γ2te−i(ǫr−ǫg)t(q−i)2, Ref.[21] including the phase shift and q parameter rela-
tionship. In the following numerical study of the phase
giving the transition stength a delta function response, shift, we always ignore the ground state energy, i.e., set-
which comes from the non-zero background cross sec- ting ǫ =0 [38].
g
tion σ , followed by a decaying single mode oscillation. The physically measurable dipole response function
0
The complex factor (q −i)2 can be cast into exponen- Eq.(14) is understood to be exact when the integral is
tial representation, (q−i)2 = (q2 +1)exp[iφ(q)], where ranging over the whole range of the energy spectrum, in
φ(q)=2arg(q−i) induces a phase shift. which case the dipole response would be a complicated
4
offset between the initial time of the system evolution
and the control pulse when interaction is turned on. To
see the effect ofthe secondpulse, we apply the evolution
operator to the initial state (6):
|ψ′(t=0)i=e−iβeiφDˆ|ψ(t=0)i
≈(1−iβeiφDˆ)(1−iγdˆ)|gi (16)
=|gi−iγ(dˆ−iβeiφDˆdˆ)|gi,
which, compared with state (6), has an overall effect of
modifying the original transition dipole operator dˆ. For
appropriate tuned phase shift, e.g., φ=±π/2, the effec-
tive transition dipole becomes
FIG. 4. Phase shift versus q parameters. Dots are values dˆ→d (1±βd )|1ih0|+d (1±βd )|2ih0|, (17)
1 1 2 2
read out from the explicit numerical calculation of the time
dependentdipole. Thesolidlineistheq-phaserelationofEq. or
(1).
d d
1 1
→ (1+β(d ∓d )). (18)
1 2
d d
2 2
mixture of different frequencies. When one is interested
Combined with the fact that the Fano line shape pa-
inobservingtheeffectofanisolatedresonanceatposition
rameter is proportional to the ratio between d to d , it
2 1
ǫ , namely, the particular frequency mode (ǫ −ǫ )/2π
r r g is concluded that the control pulse will lead to an effec-
ofthe dipole response,the integralcanbe restrictedto a
tivechangeofq-parameter,namely,amodificationofthe
finiterangeofafewresonancewidthsneartheresonance
Fano line shape.
energy, e.g., [ǫ −∆E,ǫ +∆E]. However, in this man-
r r
nerthedipoleresponsewouldbedependentonthechoice
of ∆E - the integral in Eq.(14) does not converge with IV. CONCLUSION
∆E because of the non-zero background cross section.
To eliminate this dependence without changing the crit-
To summarize, we have investigated the Fano res-
ical phase information, in the following numerical study
onance with an analytic solvable coupled-square-well
we compute the dipole response using the shifted cross
model in both frequency and time domain. The Fano
section σ(ǫ)−σ . In Fig. 3, the time-dependent dipole
0 asymmetric parameter q and the phase shift φ of the
responsefordifferentFanoq-parametersareplotted. The
magnetic dipole transition were shown to have a simple
phase shifts arereadoutandcomparedwith the phase-q
relation ψ = 2arg(q − i), which generalizes the result
correspondence Eq. (1) in Fig. (4).
originallydiscoveredin Ref.[21] for electric dipoles. This
The aboveanalysisshowsthe decayofdipole response
relation was also proven to be valid for any transition
and the change of line shapes corresponding to different
dipole, as long as an isolated Fano resonance is present
transition parameters,i.e., tuning of the internal system
in the perturbative limit.
parameter. On the other hand, it is more interesting to
show that for a system with fixed internal parameters,
resonance profiles can be modified through controlling
ACKNOWLEDGMENTS
withexternalfield. SimilartothetreatmentinRef. [21],
we introduce a subsequent control pulse, applied to the
This work was supported by the U.S. Depart-
systemimmediatelyafterthefirstexcitationpulse,mod-
ment of Energy, Office of Science, under Award No.
eled by the following interaction:
de-sc0010545.
Hˆ =βeiφδ(t)Dˆ, (15)
2
where Dˆ =d |1ih1|+d |2ih2| is the magnetic dipole mo- Appendix A: Solution of the model
1 2
ment or more generally the transitionoperator. The fre-
quencyofthe secondpulseisassumedtobe farfromres- Wepresentherethederivationofthesolutionoftheen-
onance such that excitation of the ground state channel ergy eigenfunctions and discuss the scattering behavior
by the control pulse would not take place. In the region of the coupled two square well model. The time inde-
thatthecontrolpulseismuchshorterthanthelifetimeof pendent Schrodinger equation of Hamiltonian (2) would
the system, we treat the pulse as a δ function, with two possess 4 independent solutions in general, but when we
free parameters left to be tuned: the strength β charac- restrict the solutions to obey the regular physical con-
terizingthe overalleffectofthe intensityanddurationof strains ψ(r = 0) = 0, only two linearly independent so-
thepulse,andthephaseshiftφ,characterizingthephase lutions remain. Our solution strategy will be to begin
5
by solving for these two linearly-independent solutions
that are regular at the origin. In a second step, we will Here,fornotationalconvenience,wehavedefinedavector
impose long-range boundary conditions, enforcing expo-
nential delay in the closed channel, and determine the e−ikr
~a(r)= (A5)
exact S-matrix for this model so we can study its poles (cid:18) 0 (cid:19)
inthecomplexenergyplane. Sincethecouplingpotential
and a diagonal matrix
is constant within the reaction volume, it can be diago-
nalized by an r-independent eigenvector matrix, which
eikr 0
reducesthe solutionto twouncoupledshort-rangeeigen- Dˆ(r)= . (A6)
(cid:18) 0 e−qr (cid:19)
channels. First, define the matrix
The next step is to eliminate ~z =−yˆ(r )−1XˆT~a(r )+
0 0
Wˆ = −2m Vˆ +Eˆth−εIˆ yˆ(r0)−1XˆTDˆ(r0)~s, and insert it into the derivative con-
(cid:18) ¯h2 (cid:19)(cid:16) (cid:17) tinuity equation, giving
(A1)
2m ε+V −V
= 1 12 ,
(cid:18)¯h2 (cid:19)(cid:18) −V12 ε−E2th+V2 (cid:19) Xˆyˆ′(r0)[−yˆ(r0)−1XˆT~a(r0)+yˆ(r0)−1XˆTDˆ(r0)~s]
=−~a′(r )+Dˆ′(r )~s.
and indicate the constant orthogonal eigenvector ma- 0 0
(A7)
trix as X and the (weakly) energy-dependent diago-
iα
Wigner’sreal,symmetricR-matrixisnowevidentinthis
nal eigenvalue matrix by w (ε)2. Thus we have, in ma-
α
equation, and it will simplify our algebra if we denote it
trix notation, WˆXˆ = Xˆwˆ2. Next we replace the solu-
explicitly:
tion matrix uˆ(r) by XˆXˆTuˆ(r) just before the solution
matrix uˆ(r) in the time-independent Schrodinger equa-
tion uˆ′′(r) + Wˆuˆ(r) = 0. Upon left-multiplying the Rˆ =Xˆyˆ′(r )yˆ(r )−1XˆT
0 0
whole equation by XˆT, we obtain two uncoupled single- w cotw r 0 (A8)
channel equations in the eigenrepresentation. The diag- =Xˆ 1 1 0 XˆT.
(cid:18) 0 w2cotw2r0 (cid:19)
onal eigensolution matrix at short range will be denoted
yˆ(r)=XˆTuˆ(r),andthecomponentsofthissolutionobey The above equation now reads −Rˆ~a(r )+RˆDˆ(r )~s =
the 2nd order equation, y′′(r)+w 2y (r)=0. The reg- 0 0
α α α −~a′(r )+Dˆ′(r )~s. Thus we obtain our solution for the
ular solution at the origin is of course sin(w r). 0 0
α
physically important quantities contained in~s:
Thenextstepconsistsofmatchingthis solutiontothe
simple trigonometricsolutionsthat apply outside the re-
action volume, at r > r , and imposing the physically −1
0 ~s= Dˆ′(r )−RˆDˆ(r ) ~a′(r )−Rˆ~a(r ) . (A9)
relevant boundary conditions at r → ∞. The correct 0 0 0 0
(cid:16) (cid:17) (cid:16) (cid:17)
physical solution at all distances r > r is of course a
0
More explicitly,
scatteringsolutioninthe openchannel |1i, andanexpo-
nentially decaying solution in the closed channel |2i:
eikrS−e−ikr S = (ik−R11)eikr0 −R12e−qr0 −1
ψ~phys(r)=(cid:18) Ne−qr (cid:19). (A2) (cid:18)N (cid:19) (cid:18) −R21eikr0 (−q−R22)e−qr0 (cid:19)
−ik−R
11 e−ikr0
HereS isthedesiredscatteringmatrixatenergyε,while (cid:18) −R21 (cid:19)
N is a closed-channelamplitude, which we organize into (A10)
S
a column vector ~s = . The components of this
(cid:18)N (cid:19) Now the scattering matrix S is readily evaluated, but
vector will be determined by matching this form for the instead of giving that explicit formula here, we give in-
outerregionsolutionandderivativetoourshortrangeso- stead the formula for the poles of S. These occur at
lution derivedabove,at r =r . Note that k2 =2mε/¯h2, energies for which
0
while q2 = 2m(Eth −ε)/¯h2. Neither of our two short-
2
rangeeigensolutionswillingeneralmatchsmoothlyonto
(q+R )(−ik+R )−R2 =0. (A11)
thisdesiredlongrangebehavior. We mustsuperposethe 22 11 12
two solutions with constant coefficients ~z = {z ,z }T in
1 2 This equation could now be solved numerically to de-
ordertoaccomplishthis. Thisleadstoasetofcontinuity
termine the pole positions in the complex energy plane.
equations with the structure:
However,it will be consistent with the other approxima-
tions we have made to this point if we make a linear ex-
ψ~phys(r )=Xˆyˆ(r )~z =−~a(r )+Dˆ(r )~s (A3)
0 0 0 0 pansion of q about zero energy and about the magnetic
ψ~phys′(r )=Xˆyˆ′(r )~z =−~a′(r )+Dˆ′(r )~s. (A4) field point B at which a new bound state appears or
0 0 0 0 0
6
disappears. Our approximate treatment will neglect the It may be interesting to contrast this expression with
energy and field dependences of the R-matrix itself, and the exact single-channel result for a short-range poten-
assume that the closed channel wavenumber q depends tial. The most general S-matrix for the single channel
on energy as is evident in its definition above, and on problem has the form:
magneticfieldthroughanassumedvariationoftheupper
thresholdenergywith magneticfield, i.e. E2th =E2th(B), S =e−2ikr0R(ǫ)+ik. (A17)
whereby we can write R(ǫ)−ik
Hereagain,themostgeneralenergy-dependenceforR(ǫ)
q(ǫ,B)≃q +ζk2+γ(B−B ). (A12) is a meromorphic function with poles on the real energy
0 0
axis.
Here the two real constants ζ and γ are defined by
¯h2 ∂q(ǫ,B)
Appendix B: Physical scattering length
ζ ≡ [ ] ,
2m ∂ǫ ǫ=0,B=B0
(A13)
∂q(ǫ,B)
The physical scattering length of the system at zero
γ ≡[ ] .
∂B ǫ=0,B=B0 energycanbe extractedfrom the low-energybehaviorof
the scattering phase shift. First, however, it is useful to
Three pole locations now emerge as the roots of a cubic
define the (weakly) energy-dependent scattering length,
equationin k,at anychosenfieldvalue B. The factthat
in terms of the exact S-wave scattering phase shift:
the scattering length at k = 0 is infinite when B = B
0
impliesthatq isfixedtohavethevalueq =−(R R −
0 0 11 22 tanδ(ǫ,B)
R2 )/R , which brings our final cubic equation to the a(ǫ,B)≡− . (B1)
12 11 k
form:
Thezero-energyscatteringlengthistypicallyusedinthe
ik R122 +(R −ik)(γB′ +k2)=0. (A14) context of BECs and DFGs, which is of course just the
11
ζR ζ
11 zero energy limit of this last expression, or in terms of
the scattering amplitude derived earlier,
Interestingly,thereare3realparametersthatcontrolthe
structure of these S-matrix poles in the complex energy
1
plane, namely R ,R , and γ/ζ. Each of these can be a(0,B)= lim(− lnS(ǫ,B)). (B2)
12 11 ǫ→0 2ik
assigneda direct physicalinterpretationin this problem.
Firstofall,R11canbeapproximatelyassociatedwiththe This gives the following for the zero-energy scattering
background scattering length, i.e. Y ≡ R11 ≃ −1/Abg, length as a function of magnetic field:
provided A ≫ r , as is usually the case for the atom-
bg 0
atom s-wave scattering in most alkali systems. Notice R2 +γR (B−B )
a(0,B)≃ 12 11 0 +r
thatZ ≡ h¯2γ is the slopeofthe Feshbachresonance,i.e. −γR2 (B−B ) 0
2mζ 11 0 (B3)
the variation of the resonance energy per unit change of ∆
≡a (1− ).
themagneticfield. Finally,theparameterX ≃R122/ζR11 bg B−B0
is a measure of the coupling strength between the chan-
nels, giving This expression is valid only at zero energy, and over
the range of magnetic field values for which q can be
ikX+(Y −ik)(ZB′ +k2)=0. (A15) expanded linearly in B ≈ B . The next important cor-
0
rectiontermshouldbeincludedwhenB ≈B ,wherethe
0
Theactualscatteringamplitudeitselftakesthefollow- denominatorshouldincludealinearfunctionofenergyin
ing form, in terms of the original R-matrix elements: order to obtain a more general and effective parameteri-
zation, i.e.
R2 −R R −qR −ik(q+R )
S =e−2ikr0 12 11 22 11 22 .
R2 −R R −qR +ik(q+R ) ∆
12 11 22 11 22 a(ǫ,B)≃a (1− ). (B4)
(A16) bg B−B +ζǫ
0
In thinking about the energy dependence of this scat-
teringmatrix,itshouldberememberedthateachelement This form for the general phase shift is very accurate,
of the R-matrix is in general a meromorphic function of typicallywithinapproximately1to10microkelvinabove
the energy. However,sincethe scaleofshort-rangeinter- and below zero energy.
actions is typically huge compared to the ultra-cold en-
ergyscale,it willusually be a goodapproximationto re-
gardeachelementoftheR-matrixasenergy-independent Appendix C: Bound State Energy Level Properties
in applications at sub-microkelvintemperatures. Also, a
linear expansion of q as a function of ǫ and B can be The above wavefunction used to describe low energy
inserted, as was discussed above. atom-atom scattering still applies at negative energies,
7
ǫ = −h¯2κ2. For definiteness, I assume that the analytic simply by:
2m
continuation in going from positive to negative energies
is carried out by setting k → iκ, with the convention −γR2 (B−B )
κ= 11 0 . (C3)
that κ is a real, positive number in this regime. Then R2 +γR (B−B )
the entire derivation could be repeated from the begin- 12 11 0
ning, of course, but a shorter route to the desired result
Since the bound state energy is ǫ= −h¯2κ2 (provided the
just begins from the above unnormalized wavefunction, 2m
preceding expression for κ is positive), this proves that
except we divide it by S(ǫ,B) . The wavefunctionin the
thebindingenergyofahigh-lyingboundlevelalwaysap-
?weakly-closed? channel is then
proaches 0 quadratically in the magnetic field, except in
ψ →e−κr−eκrS−1, (C1) the uninteresting limit where the channels are noninter-
acting.
which will be unphysical and diverge exponentially un- Anotherquantityofphysicalinterestistheprobability
less S−1 → 0 for some κ > 0. Referring to the above that the system resides in the upper (strongly-closed)
expression for S, the condition for a bound state thus channel. In the limit of a zero-range potential r → 0,
0
becomes: and in the limit of very small binding where κ → 1/A,
this probability is given by
R2 −R R −qR
κ= 12 11 22 11. (C2)
q+R
22 (1+AR )2
11
Probability(|2i)≃ , (C4)
Thelinearexpansioncannowbeinserted,q ≃q0+γ(B− (1+AR11)2+A3q0R122
B ), i.e. neglecting the weak energy dependence of q.
0
Whenthisresultiscombinedwiththefactthatthepoint which vanishes as 1/A in the limit where the physical
atwhichthescatteringlengthisinfinitehasbeendefined scattering length diverges, i.e. when B → B and A →
0
tobeB ,theboundstatewavenumberisseentobegiven ∞.
0
[1] S.R. Leone, C. W. McCurdy, J. Burgdrfer, L. S. Ceder- Z. Chang, Phys.Rev. Lett.105, 263003 (2010).
baum, Z. Chang, N. Dudovich, J. Feist, C. H. Greene, [12] W.-C. Chu, T. Morishita, and C. D. Lin, Phys. Rev. A
M. Ivanov, R. Kienberger, U. Keller, M. F. Kling, Z.-H. 89, 003427 (2014).
Loh, T. Pfeifer, A. N. Pfeiffer, R. Santra, K. Schafer, [13] J. Mauritsson, T. Remetter, M. Swoboda, K. Klunder,
A. Stolow, U. Thumm, and M. J. J. Vrakking, Nature A. LHuillier, K. J. Schafer, O. Ghafur, F. Kelkensberg,
Photonics 8, 162 (2014). W. Siu, P. Johnsson, M. J. J. Vrakking, I. Znakovskaya,
[2] A.Kaldun, A. Blttermann, V. Stooβ, S.Donsa, H. Wei, T. Uphues, S. Zherebtsov, M. F. Kling, F. Lepine,
R.Pazourek, S.Nagele, C. Ott,C. D. Lin, J. Burgdrfer, E. Benedetti, F. Ferrari, G. Sansone, and M. Nisoli,
and T. Pfeifer, Science 354, 738 (2016). Phys. Rev.Lett. 105, 053001 (2010).
[3] V. Gruson, L. Barreau, A. Jimnez-Galan, F. Risoud, [14] C. Ott, A. Kaldun, L. Argenti, P. Raith, K. Meyer,
J. Caillat, A. Maquet, B. Carre, F. Lepetit, J. F. Her- M. Laux, Y. Zhang, A. Blattermann, S. Hagstotz,
gott, T. Ruchon, L. Argenti, R. Taieb, F. Martin, and T. Ding, R. Heck, J. Madronero, F. Martin, and
P.Salieres, Science 354, 734 (2016). T. Pfeifer, Nature516, 7531 (2013).
[4] M. Kotur, D. Gunot, . Jimnez-Galn, D. Kroon, E. W. [15] J. M. Lecomte, A. Kirrander, and C. Jungen, J. Chem.
Larsen, M. Louisy, S. Bengtsson, M. Miranda, J. Mau- Phys. 139, 164111 (2013).
ritsson, C. L. Arnold, S. E. Canton, M. Gisselbrecht, [16] L.Argenti,R.Pazourek,J.Feist,S.Nagele, M.Liertzer,
T. Carette, J. M. Dahlstrm, E. Lindroth, A. Maquet, E. Persson, J. Burgdorfer, and E. Lindroth, Phys. Rev.
L.Argenti,F.Martn, andA.LHuillier,NatureCommu- A 87, 053405 (2013).
nications 7 (2016). [17] A.Maquet,J.Caillat, andR.Taieb,J.Phys.B:At.Mol.
[5] K. Ramasesha, S. R. Leone, and D. M. Neumark, Ann. Opt. Phys.47, 204004 (2014).
Rev.Phys.Chem. 67, 41 (2016). [18] L. Mediauskas, F. Morales, A. Palacios, A. Gonzalez-
[6] A.Maquet,J.Caillat, andR.Taeb,J.Phys.B:At.Mol. Castrillo, L.Plimak,O.Smirnova,F.Martn, andM.Y.
Opt.Phys.47 (2014). Ivanov,New J. Phys. 17, 053415 (2015).
[7] A.R.Becka,D.M.Neumarka, andS.R.Leonea,Chem. [19] W. C. Chu and C. D. Lin, Phys. Rev. A 82, 053415
Phys.Lett 624, 119130 (2015). (2010).
[8] W.-C. Chu, T. Morishita, and C. D. Lin, Phys. Rev. A [20] M. Wickenhauser, J. Burgdorfer, F. Krausz, and
89, 033427 (2014). M. Drescher, Phys.Rev.Lett. 94, 023002 (2005).
[9] Z.Q.Yang,D.F.Ye,T.Ding,T.Pfeifer, andL.B.Fu, [21] C. Ott,A.Kaldun,P.Raith,K. Meyer,M. Laux,J. Ev-
Phys.Rev.A 91, 013414 (2015). ers, C. H. Keitel, C. H. Greene, and T. Pfeifer, Science
[10] H. Wang, M. Chini, S. Chen, C.-H. Zhang, F. He, 340, 716 (2013).
Y. Cheng, Y. Wu, U. Thumm, and Z. Chang, Phys. [22] U. Fano, Phys. Rev.124, 1866 (1961).
Rev.Lett. 105, 143002 (2010). [23] A. Hibbert,Rep. Prog. Phys.38, 1217 (1975).
[11] S. Gilbertson, M. Chini, X. Feng, S. Khan, Y. Wu, and [24] P. E. Grabowski and D. F. Chernoff, Phys. Rev. A 84,
8
042505 (2011). Mod. Phys.82, 1225 (2010).
[25] A.Kaldun,C.Ott,A.Bl?ttermann,M.Laux,K.Meyer, [33] H. Bethe, Phys. Rev.47, 747 (1935).
T. Ding, A. Fischer, and T. Pfeifer, Phys. Rev. Lett. [34] S.J.J.M.F.Kokkelmans,J.N.Milstein,M.L.Chiofalo,
112, 103001 (2014). R.Walser, andM.J.Holland,Phys.Rev.A65,053617
[26] S. V. Dordevic1, G. M. Foster1, M. S. Wolf1, N. Sto- (2002).
jilovic, H.Lei, C.Petrovic,Z.Chen,Z.Q.Li, andL.C. [35] R. A. Duine and H. T. C. Stoof, Phys. Rep. 396, 115
Tung,J. Phys.: Condens. Matter 28 (2016). (2004).
[27] S. Yoshino, G. Oohata, and K. Mizoguchi, Phys. Rev. [36] A. D. Lange, K. Pilch, A. Prantner, F. Ferlaino, B. En-
Lett 115, 157402 (2015). geser, H.-C. Naegerl, R. Grimm, and C. Chin, Phys.
[28] R. A. Shah, N. F. Scherer, M. Pelton, and S. K. Gray, Rev. A 79, 013622 (2009).
Phys.Rev.B 88, 075411 (2013). [37] U. Fano and J. W. Cooper, Rev. Mod. Phys. 40, 441
[29] J. Wallauer and M. Walther, Phys. Rev. B 88, 195118 (1968).
(2013). [38] Strictly speaking, the ground state energy cannot be
[30] D.-Y.Chen,X.Liu,X.-Q.Li, andH.-W.Ke,Phys.Rev. zero as the lower energy threshold has already been de-
D 93, 014011 (2016). fined to be zero. However, since the ground state energy
[31] M. J. Akram, M. M. Khan, and F. Saif, Phys. Rev. A onlychangestheoveralloscillationfrequencybutnotthe
92, 023846 (2015). phase information, for simplicity we compute the dipole
[32] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. responsewithǫg =0,whichisequivalentaschangingthe
time scale to (ǫr−ǫg)/ǫr.